---
tag: 0K5W69AC0J
kind: object
name: Gaussian integers
msc: [11R11]
instance_of: [0CHS7RBD39]
uses: [0859D58WPD, 09S3PFBEPM]
sources:
  - url: https://en.wikipedia.org/wiki/Gaussian_integer
    locator: Basic definitions
same_as:
  wikidata: Q724975
  wikipedia: Gaussian integer
---

## Description

The ring of Gaussian integers is
$$\mathbb{Z}[i] = \{a + b i : a, b \in \mathbb{Z}\},$$
where $i \in \mathbb{C}$ satisfies $i^2 = -1$.
It is a [[subring|0859D58WPD]] of $\mathbb{C}$, with $(a + b i)(c + d i) = (a c - b d) + (a d + b c) i$.
As a subring of a field, it is an [[integral domain|09S3PFBEPM]].

## Remarks

The [[norm|03AN1V4ZFJ]] $N(a + b i) = a^2 + b^2$ is multiplicative.
The [[units of $\mathbb{Z}[i]$|06RH4XAGRY]] are $1, -1, i, -i$.
The ring $\mathbb{Z}[i]$ is a [[Euclidean domain|0PN13BB7BT]], so it has unique factorization.
A prime number $p$ is a sum of two squares if and only if $p$ is not prime in $\mathbb{Z}[i]$. This holds exactly for $p = 2$ and the primes $p \equiv 1 \pmod 4$.

## Explanations

### L1 human

The Gaussian integers are the points with integer coordinates in the complex plane.
You add and multiply them as complex numbers.
They form a ring with a theory of primes and factorization, like the integers.